MHB Probability of Consecutive Data Points Rising and Falling in a Saw-tooth Pattern

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The discussion focuses on calculating the probability of consecutive data points forming a saw-tooth pattern, defined as alternating mountains and valleys. Each set of three consecutive points must create either a mountain or a valley, with the magnitudes of these formations being irrelevant. The probability of a point being higher or lower than the previous one is equal, leading to the formulation of a difference equation for the probability of the saw-tooth pattern. The derived solution indicates that the probability for n points is given by p_n = 2^(2-n), with p_2 set to 1. This mathematical approach provides a clear framework for understanding the occurrence of saw-tooth patterns in uniformly distributed random variables.
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What is the correct way to calculate the probability of a given number of consecutive data points forming a saw-tooth pattern?
The magnitudes of the rise or fall (the size of the moutain and valleys) are not material.
The only requirement is that each set of three consecutive data points must form either a mountain or a valley, and that there is 1 mountain, followed by 1 valley, followed by 1 mountain, followed by 1 valley, etc.
 
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Let's suppose to have n random variable $\displaystyle x_{i}\ , i=1,2,...,n$ each of them uniformly distributed from 0 to 1. In this situation You have...

$\displaystyle P\{x_{i+1}<x_{i}\}=P\{x_{i+1}>x_{i}\}=\frac{1}{2}$ (1)

Let's set with $p_{n}$ the probability that the n $x_{i}$ form a 'saw tooth pattern'. It is quite obvious that $p_{2}=1$ so that , if the (1) is true, $p_{n}$ is the solution of the difference equation...

$\displaystyle p_{n+1}=\frac{p_{n}}{2}\ ,\ p_{2}=1$ (1)

... and that solution is $\displaystyle p_{n}=2^{2-n}$...

Kind regards

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